[FOM] Deflationism and the Godel phenomena

[Aatu Koskensilta]

On Feb 17, 2005, at 2:51 PM, Joseph Vidal-Rosset wrote:

> Aatu Koskensilta a écrit :
> |
> | On Feb 16, 2005, at 12:58 PM, Joseph Vidal-Rosset wrote:
> |
> |> Do we need at this point to grow up to the second order? I believe 
> it.
> |
> | Only if we want to be able to define the truth predicate explicitly.
>
> Thanks.
> But this logico-philosophical debate around deflationism about truth
> tries to define truth explicitly, wondering after its substantial or
> non-substantial property.

It's perfectly acceptable to define the truth predicate implicitly by
listing the inductive clauses it has to satisfy. A basic fact about 
first
order truth is that there is a unique predicate (set) satisfying these
clauses.

In case you're wondering, the clauses look something like this:

  True('A&B')              <=> True('A') & True('B')
  True('~A')               <=> ~True('A')
  True('P_1(t_1,...,t_n)') <=> P_1(t_1,...,t_n)
                            .
                            .
                            .
  True('P_i(t_1,...,t_n)') <=> P_i(t_1,...,t_n)
  True('ExA')              <=> EyTrue('A'[x/y])

(where P_i are the predicate symbols of the language in question).

For example, the theory Tr(PA) Jeffrey Kettland mentioned is obtained 
from PA by
adding to its language a predicate True, adding the above clauses as 
axioms
(with {P_i} = {+,*,=,S}) and extending the induction schema to cover 
all formulae
of the extended language. The resulting theory is not conservative over 
PA,
since e.g. the trivial proof of PA's consistency can be carried out 
(the axioms
of PA are true and the rules of inference of first order logic preserve 
truth
hence no contradiction is provable from the axioms of PA).

Feferman (and others) has pointed out that the most natural 
formalization of
arithmetic in first order logic uses free predicate variables with a 
substitution
rule saying that any definable predicate can be substituted for the 
free predicate
variable. Thus it might be argued that the theory Tr(PA) is the correct 
formalization
of what one gets by adding an arithmetical truth predicate to PA and 
hence arithmetical
truth is substantial in the sense that more arithmetical truths can be 
proved by
using the concept than without it. I'm sure Jeffrey will correct me if 
I have
manhandled his argument too roughly.

> Gödel believed in truth as substantial, no doubt, but neither Tarski 
> nor
> Carnap did.

What belief exactly do you have in mind?

-- 
Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus